Working with the definitions and proving properties of restriction/pushforward/pullback of sheaves was okay until I realized that I can't do calculation of a simple example:
Let $X=\text{Spec} (k[x,y])$ and $Z=\text{Spec}(k[x,y]/(y))$ with $i:Z\rightarrow X$ being an obvious inclusion. I tried to calculate the sections of $i^{-1}O_X$ inverse presheaf(!) over distinguished open $D(x)$ but got stuck, let alone the inverse sheaf itself.
I know that $i^{-1}(D(x))=colim_{D(x)\cap V(y)\subset U}O_X(U)$. I can't take $U$ only as a distinguished open on $X$ because in our directed system distinguished subsets are not cofinal. So it just seems too cumbersome job to do because I have to glue arbitrary $D(f)$ to arbitrary $U$ and then take a colimit.
Is there smarter way to calculate this presheaf inverse or sheaf inverse? Maybe I'm missing something simple.
Maybe we can simply take $X= \Bbb A^1_k$ and $Z = \{0\}$ where everything is clearer. In this case $i^{-1}\mathcal O_X$ is given by $$\text{colim}_{0 \in U} \mathcal O_X(U)$$
An element in such a limit is exactly a function $f/g$ where $f,g \in k[x] $ and $g(0) \neq 0$. This sheaf is pretty big and in particular not coherent on $\{0\}$ since it has infinite dimension.
This is why the definition of a restriction of a sheaf is $i^* F = i^{-1}F \otimes_{i^{-1}\mathcal O_X}\mathcal O_Z$, to get again something coherent. In particular for any closed subvariety $Z \subset X$ we have $i^* \mathcal O_X = \mathcal O_Z$.
In general, the sheaf $i^{-1} \mathcal O_X$ correspond to functions on $X$ with poles outside $Z$. In your example, a typical element in your sheaf would be $\frac{1}{y+1}$.